Completeness in the theory of types

@article{Henkin1950CompletenessIT,
  title={Completeness in the theory of types},
  author={Leon Henkin},
  journal={Journal of Symbolic Logic},
  year={1950},
  volume={15},
  pages={81 - 91}
}
  • L. Henkin
  • Published 1 June 1950
  • Mathematics
  • Journal of Symbolic Logic
The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system. For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a… 

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References

SHOWING 1-3 OF 3 REFERENCES

Die Vollständigkeit der Axiome des logischen Funktionenkalküls

Jakina da Whiteheadek eta Russellek logika eta matematika eraiki dutela ageriko zenbait proposizio axiomatzat hartuz, eta horietatik, zehatz azaldutako inferentzia printzipioetan oinarrituz, logikako